American Institute of Mathematical Sciences

Advanced
January  2013, 12(1): 519-546. doi: 10.3934/cpaa.2013.12.519

Numerical study of a family of dissipative KdV equations

Jean-Paul Chehab 1, and  Georges Sadaka 1,

1. 

LAMFA, UMR 6140, Université de Picardie Jules Verne, Pôle Scientifique, 33, rue Saint Leu, 80039 Amiens, France, France

Received  April 2011 Revised  December 2011 Published  September 2012

The weak damped and forced Korteweg-de Vries (KdV) equation on the 1d Torus have been analyzed by Ghidaglia[8, 9], Goubet[10, 11], Rosa and Cabral [3] where asymptotic regularization e ects have been proven and observed numerically. In this work, we consider a family of dampings that can be even weaker, particularly it can dissipate very few the high frequencies. We give numerical evidences that point out dissipation of energy, regularization e ect and the presence of special solutions that characterize a non trivial dynamics (steady states, time periodic solutions).
Keywords: Korteweg-de Vries equations, long time behavior, damping.
Mathematics Subject Classification: 35B10, 35B40, 35Q53, 65M12, 65M70.
Citation: Jean-Paul Chehab, Georges Sadaka. Numerical study of a family of dissipative KdV equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 519-546. doi: 10.3934/cpaa.2013.12.519
References:
[1]

M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete schrodinger equations and dissipative dynamical systems,, Communications on Pure and Applied Analysis, 7 (2008), 211.   Google Scholar

[2]

M. Abounouh, H. Al Moatassime, C. Calgaro and J-P. Chehab, A numerical scheme for the long time simulation of a forced damped KdV equation,, in preparation., ().   Google Scholar

[3]

M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Phys. D, 192 (2004), 265.  doi: 10.1016/j.physd.2004.01.023.  Google Scholar

[4]

C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Schémas multiniveaux pour les équations d'ondes,, (French) [Multilevel schemes for waves equations], 27 (2009), 180.  doi: 10.1051/proc/2009027.  Google Scholar

[5]

J.-P. Chehab and B. Costa, Time explicit schemes and spatial finite differences splittings,, Journal of Scientific Computing, 20 (2004), 159.   Google Scholar

[6]

J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations,, preprint, (2011).   Google Scholar

[7]

D. Dutykh, Modélisation mathématique des Tsunamis,, (French) [Mathematical modeling of Tsunamis], (2007).   Google Scholar

[8]

J-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time,, J. Diff. Eq., 74 (1988), 369.  doi: 10.1016/0022-0396(88)90010-1.  Google Scholar

[9]

J-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations,, J. Diff. Eq., 110 (1994), 356.  doi: 10.1006/jdeq.1994.1071.  Google Scholar

[10]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625.   Google Scholar

[11]

O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line,, J. Differential Equations, 185 (2002), 25.  doi: 10.1006/jdeq.2001.4163.  Google Scholar

[12]

D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications,", SIAM Philadelphia, (1977).  doi: 10.1115/1.3424477.  Google Scholar

[13]

S. Mallat, "A Wavelet Tour of Signal Processing,", Academic press, (1998).   Google Scholar

[14]

A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics,", Cambridge University Press, (2005).   Google Scholar

[15]

L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case,, preprint, ().   Google Scholar

[16]

E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping,, Physics of fluids, 12 (1969), 2388.  doi: 10.1063/1.1692358.  Google Scholar

[17]

E. Ott and R. N. Sudan, Damping of solitary waves,, Physics of fluids, 13 (1970), 1432.  doi: 10.1063/1.1693097.  Google Scholar

[18]

A. Duràn and J. M. Sanz-Serna, The numerical integration of a relative equilibrium solutions. the nonlinear schrodinger equation,, IMA J. Num. Anal., 20 (2000), 235.  doi: 10.1093/imanum/20.2.235.  Google Scholar

[19]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", 2$^{nd}$ edition, (1997).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[20]

S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations,, Funkcialaj Ekvacioj, 54 (2011), 119.  doi: 10.1619/fesi.54.119.  Google Scholar

[21]

S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations,, Asymptot. Anal., 68 (2010), 155.   Google Scholar

show all references

References:
[1]

M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete schrodinger equations and dissipative dynamical systems,, Communications on Pure and Applied Analysis, 7 (2008), 211.   Google Scholar

[2]

M. Abounouh, H. Al Moatassime, C. Calgaro and J-P. Chehab, A numerical scheme for the long time simulation of a forced damped KdV equation,, in preparation., ().   Google Scholar

[3]

M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Phys. D, 192 (2004), 265.  doi: 10.1016/j.physd.2004.01.023.  Google Scholar

[4]

C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Schémas multiniveaux pour les équations d'ondes,, (French) [Multilevel schemes for waves equations], 27 (2009), 180.  doi: 10.1051/proc/2009027.  Google Scholar

[5]

J.-P. Chehab and B. Costa, Time explicit schemes and spatial finite differences splittings,, Journal of Scientific Computing, 20 (2004), 159.   Google Scholar

[6]

J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations,, preprint, (2011).   Google Scholar

[7]

D. Dutykh, Modélisation mathématique des Tsunamis,, (French) [Mathematical modeling of Tsunamis], (2007).   Google Scholar

[8]

J-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time,, J. Diff. Eq., 74 (1988), 369.  doi: 10.1016/0022-0396(88)90010-1.  Google Scholar

[9]

J-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations,, J. Diff. Eq., 110 (1994), 356.  doi: 10.1006/jdeq.1994.1071.  Google Scholar

[10]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625.   Google Scholar

[11]

O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line,, J. Differential Equations, 185 (2002), 25.  doi: 10.1006/jdeq.2001.4163.  Google Scholar

[12]

D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications,", SIAM Philadelphia, (1977).  doi: 10.1115/1.3424477.  Google Scholar

[13]

S. Mallat, "A Wavelet Tour of Signal Processing,", Academic press, (1998).   Google Scholar

[14]

A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics,", Cambridge University Press, (2005).   Google Scholar

[15]

L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case,, preprint, ().   Google Scholar

[16]

E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping,, Physics of fluids, 12 (1969), 2388.  doi: 10.1063/1.1692358.  Google Scholar

[17]

E. Ott and R. N. Sudan, Damping of solitary waves,, Physics of fluids, 13 (1970), 1432.  doi: 10.1063/1.1693097.  Google Scholar

[18]

A. Duràn and J. M. Sanz-Serna, The numerical integration of a relative equilibrium solutions. the nonlinear schrodinger equation,, IMA J. Num. Anal., 20 (2000), 235.  doi: 10.1093/imanum/20.2.235.  Google Scholar

[19]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", 2$^{nd}$ edition, (1997).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[20]

S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations,, Funkcialaj Ekvacioj, 54 (2011), 119.  doi: 10.1619/fesi.54.119.  Google Scholar

[21]

S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations,, Asymptot. Anal., 68 (2010), 155.   Google Scholar

[1]

Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312
[2]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003
[3]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336
[4]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229
[5]

Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033
[6]

Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021007
[7]

Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322
[8]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318
[9]

Jean-Paul Chehab. Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021002
[10]

Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299
[11]

Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020360
[12]

Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053
[13]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091
[14]

Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112
[15]

Hedy Attouch, Aïcha Balhag, Zaki Chbani, Hassan Riahi. Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021010
[16]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403
[17]

Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328
[18]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571
[19]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242
[20]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences